1.1 Objectives

• In module 2, we discussed the intricacies of avoiding Type I and II errors in hypothesis testing, the connection to statistical power, and how assumption violations can create problems in Type II errors

• In this module, students should be able to:

  • Understand the different possible options to pursue in light of assumption violations
  • Appreciate the use of variable transformations in addressing problems in skew, and using trimming/winsorizing to address kurtosis, i.e., solving problems in normality
  • Understand non-parametric tests as viable alternative to our usual parametric tests, with certain nuances to be aware of in application.

1.2 Introduction

• Previously, we talked about identifying problems in 3 types of common assumptions:
  • Normality
  • Homogeneity of variances
  • Independence of sampling variables
• Now we will go through the Solutions for Assumption Violations, and how they may (or may not) solve our problems
  • Our data is rarely ideal! We might need to readily consider these options

• Discuss: Of the top of your head, try to remember the different methods of identifying the above assumption violations

2.1 3 General Strategies

• Important: There is considerable controversy and conflicting ideas in this area, as to 'what should we do' - take non-parametric class for more fun!

• We need a way to address assumption violations when they occur (well mostly - see Do Nothing!)
  • As mentioned before, the main risk of assumption violations, is that they reduce power and raise Type II error rate
  • Since we are coming into most studies with a hypothesis of differences existing, we want to plan an analysis that has the sufficient power to detect that hypothesized difference
• We have 3 strategies for approaching assumption violations:
  • Do Nothing!
  • Correct for Violations in Data
  • Use Non-parametric Tests
• Before we go through those, lets talk about some reasonable advice for working through these:

2.2 Advice in ‘Fixing’ Assumption Violations

• First, always report checks/tests for assumptions of the test you are using
  • E.g., normality tests like Kolmogorov-Smirnov and Shapiro-Wilk; histograms to examine skew and kurtosis; Levene’s test for homogeneity of variances, etc.
  • Review notes from module 2 for more details on each of those
• Second, you should report assumption checks both before AND after you attempt a fix
  • I.e., If you transform/trim a variable, you should check assumption both before AND after the transformation/trim
  • Re-check ALL assumptions, not just the ones you attempted to fix with the transformation or trimming
• Third, be honest and transparent when it seems like a solution does not work or when the downsides outweigh the results
  • We’ll talk about different pitfalls when using each strategy
  • It’s good practice to always be straightforward about limitations of our analyses

• Important: The emphasis is always on transparently reporting results!

2.3 Do Nothing!

• Without going into too much detail, we are concerned with how robust a test is, or how resilient a test is to assumption violations, and how well it works under less-than-ideal circumstances

• Some researcher’s hold that many commonly used tests, i.e., t-tests are reasonably robust at baseline to assumption violations

  • This becomes even more true at large \(n\) size (See discussion in Sample Size section in module 2 notes)
  • However, when samples are small or the groups being compared are unequal - this route cannot be recommended

• Discuss: How often do you think this option is taken in 'real' research?

3.1 Transformations / Dealing with Skewness

• There are several options for selectively transforming and or trimming our data to correct for certain patterns of skewness, kurtosis, or outliers

  • However, be aware these are not panaceas - they come with their own issues

• Making mathematical variable transformations is largely used to address the Normality Assumption, but maybe indirectly solve other issues as well

• The exact transformation is dependent on the type of problem, specifically the skew:

  • Positive/right skew: Logarithmic (Severe) or Square Root (Moderate)
    • Logarithmic method: \(\log_{10}(X)\)
    • Square root method: \(\sqrt{X}\)
    • \(X\) is the variable of interest
  • Negative/left skew: Reflect and Logarithmic (Severe) or Reflect and Square Root (Moderate)
    • Logarithmic method: \(\log_{10}(K - X)\)
    • Square root method: \(\sqrt{K - X}\)
    • \(K\) is a constant from which each score is subtracted, so that the smallest score is equal to 1

• Advantages:

  • Makes use of all available data
  • Allows for use of a traditional well-known technique.

• Disadvantages:

  • Interpretability can be questionable
  • Fixing one assumption violation can create others

• Question: I have a variable of numeric test scores, in which a histogram shows that there is severe bunching of scores to the left, with a tail off to the right. Which of the above transformation might be the best to try first?
• A) Log
• B) Square Root
• C) Reflect and Log
• D) Reflect and Square Root

3.2 Trimming and Winsorizing / Dealing with Kurtosis

• Another option, particularly useful for negatively kurtotic (platykurtic) distributions (relatively flat distributions with an unusual number of observations in the tails) is to use variable trimming.

• A trimmed sample is a sample where a fixed percentage of extreme values is removed from each tail.

  • Of course, if you are comparing groups, you would want to trim the same percentage from the tails of both groups to be fair.
  • 20 percent or 0.10 from each tail is the most common amount to trim
  • Example Mean: \(((6.0 + 8.1 + 8.3 + 9.1 + 9.9) / 5) = 8.28\)
  • Example 20% Trimmed Mean: \((8.1 + 8.3 + 9.1) / 3 = 8.50\)

• Another related option is using winsorizing

  • A Winsorized sample replaces the trimmed values by the most extreme value remaining in each tail.
  • Example Dataset: 2, 4, 7, 8, 11, 14, 18, 23, 23, 27, 35, 40, 49, 50, 55, 60, 61, 61, 62, 75
  • Example 20% Winsorized Dataset: 7, 7, 7, 8, 11, 14, 18, 23, 23, 27, 35, 40, 49, 50, 55, 60, 61, 61, 61, 61
  • Note: degrees of freedom (df) for a test on a trimmed or Winsorized sample must be adjusted for the data trimming. For both, you would subtract your total N by the number of trimmed cases to get a new value for N. (We do this even for Winsorized samples b/c the added values are really pseudovalues.)

• Advantages:

  • Allows for use of a traditional well-known technique.
  • Interpretability of variable remains intact

• Disadvantages:

  • Loss of information - what if those outliers were an important part of the phenomenon

• Discuss: Using the variable data 1, 2, 3, 4, 5, try 20 percent trimming and winsorizing this data

4.1 Introduction

• Non-parametric test are distribution -free statistical tests that are not based on parameters (i.e., means or standard deviations) or assumptions about the normality underlying data distribution.

• They do still have some assumptions, just not about normality! More on that in Assumptions of Non-parametric Tests

• Instead, non-parametric tests are based on amounts such as percentages (Chi-square) or ranks, i.e., ordinal or ordinal -transformed data

• Non-parametric tests are just as powerful as traditional tests, and under situations of violated assumptions can be much more powerful

• We’ll discuss the Wilcox Rank Sum Test and the Mann Whitney U Test as they are non- parametric tests comparing the rank orderings of two independent samples

  • These can be thought of as the non-parametric versions of the independent samples t-test

• We’ll also cover the Wilcox Matched Pairs Signed Rank Test, sort of an analog to the dependent-samples t-test

• Because these test use ranks, they can actually use ordinal data, unlike the t-test!

• Discuss: Think of a hypothesis/scenario in which you'd use the independent samples t-test

4.2 Wilcox Rank Sum Test

• The Wilcox Rank Sum Test is based on the logic that if there truly is a significant difference between two groups, the ranks from one group should generally be lower than the ranks from the other group.

• Following from that, if the groups are different, the sum of one group should be lower than the sum of the ranks from the other group.

  • If the sum of the one group is too small relative to the other sum we will reject the null hypothesis.

Calculating Test Statistic

• First, combine both groups and rank their individual values from smallest (starting at 1) to largest
  • In the scenario of ties, you can assign them all the mean rank or assign adjacent ranks at random

• Second, calculate the sum of the ranks of the smaller group, also called \(W_s\).
  • The “smaller” group is the one with a smaller \(n\)

• Third, compare \(W_s\) against critical value table, which is derived from sample sizes from the smaller and larger groups
  • Realistically, we’ll use SPSS for this

• Discuss: Try calculating the rank sums for groups of data: A: 23, 34, 89; B: 16, 12, 40, 50; which of these is $W_s$?

• Important: Contrary to what we are used to, we want our test statistic to be *less* than the critical value

• This is a one -tailed test that tests to see if the ranks of the smaller group are sufficiently smaller than the larger group.

When Smaller Group Has Larger Ranks

• A problem we may run into in the above process is if the smaller group (by \(n\)) actually has the larger ranks
  • To solve for this, we can backwards rank the data, i.e., start with the largest value as 1
• We can check for that possibility by calculating \(W_s'\) or the compliment of \(W_s\)
  • \(W_s' = 2\bar{W} - W_s\) where
• Where:
  • \(2\bar{W} = n_1(n_1 + n_2 + 1)\)
  • \(n_1\): sample size of smaller sample
  • \(n_2\): sample size of the larger sample

• Important: So effectively, we have to consider the statistic both ways to account for both directions. This gets covered by our next test.

4.3 Mann Whitney U Test

• The Mann Whitney U Test is a test completely equivalent to the Wilcoxon Rank Sum, however, it is slightly more complex in order to eliminate the need to compute \(W_s'\).

• Important: For this reason, it tends to be a bit more immediately useful, because it doesn't require going through the equation twice.

• The Mann Whitney U, like the Wilcoxon Rank Sum, is based on ranked data.

Calculating Test Statistic

• We start with the exact same steps as the Wilcox Rank Sum Test, calculating \(W_s\)

• The we calculate the \(U\) statistic:

\[ U = \frac{n_1(n_1 + 2n_2 + 1)}{2} - W_s \]

• Where:
  • \(n_1\): sample size of smaller sample
  • \(n_2\): sample size of the larger sample

• Important: SPSS will only calculate Mann Whitney U, but because these tests are equivalent, that is fine!

4.4 Wilcox Matched Pairs Signed Rank Test

• The Wilcox Matched Pairs Signed Rank Test is a non- parametric test testing the null hypothesis that two related samples were drawn from identical populations with the same mean.
  • This is a non-parametric version of the dependent/paired samples t-test.
• The logic of the Signed Ranks Test rests on measuring the direction and magnitude of change.

Calculating Test Statistic

• Compute difference scores (\(d\)) between time 1 \(t_1\) and time 2 (\(t_2\))

• Next, rank regardless of sign

• There are two kinds of ties possible this time:
  • Like before, you can have rank ties which you would handle by either random ranks or tied ranks.
  • Now, though, you have the possibility for paired values to be equal (difference = 0). When this happens do not rank that case, and drop it from your sample.
• Next, Sum positive and negative ranks

• Our test statistic is \(T\), which is the smaller of the two absolute sums

4.5 Assumptions of Non-parametric Tests

• These mentioned non-parametric tests allow us to ignore normality problems in our data
  • However, we still need to test for and address problems in homogeneity of variances
  • See prior module lecture on examining histograms and Levene's test to help identify these issues

4.6 Effect Size Under Non-parametric Tests

• We can use this quick equation and rule-of-thumb for effect size under these tests

\[ r = \frac{z}{\sqrt{n}} \]

• Where:
  • \(z\) is the standardized test value
  • \(n\) is the sample size
• Interpretation
  • Small: 0.1
  • Medium: 0.2
  • Large: 0.3

• Important: Like with many statistics, there are *many* other ways of calculating effect size, some with much more complex equations - but for the scope of this class, we'll use this

4.7 Advantages and Disadvantages of Non-parametric Tests

• Advantages of Non-Parametrics
  • Interpretability of variables remains intact
  • Makes use of all available data
  • We still have good options for statistical significance testing, effect size, and other normal comforts of parametric tests
• Disadvantages
  • Non-parametric tests are relatively unknown. Many researchers would not know how to interpret them, and thus, may prefer just to see the parametric version, regardless of issues

• Discuss: Do you think this is disadvantage is a sufficient reason to avoid these tests? What issues (if any) do you have with this view?

5.1 Recap

• We have now discussed various methods we may pursue when dealing with oddities in our data that result in assumption violations

• Some of these strategies involve carefully modifying our data with transforming, trimming, or winsorizing; some strategies involve alternative selection of tests; and then finally, we may do nothing (mindfully)

• Each of our options result in some new things to be careful about, like transformation changing the scale of our data, trimming removing data, or non-parametric tests changing interpretation of results

• Our discussion of the non-parametric analogs for the t-test will help lead us into a similar discussion on alternative tests to the one-way ANOVA (and others) later in the semester

• Paramount to all of these things, we must be mindful of transparency when working through assumption problems

5.2 Lecture Check-in

• Make sure to complete any lecture check-in tasks associated with this lecture!

Module 3 Lecture - Transformations and Non-parametric Comparisons for Two Groups || Analysis of Variance